# Selection Rules

## Selection rules (physics)

General rules concerning the transitions which may occur between the states of a quantum-mechanical physical system. They derive in almost all cases from the symmetry properties of the states and of the interaction which gives rise to the transitions. The system may have a classical (nonquantum) counterpart, and in this case the selection rules may often be related to the classical conserved quantities. A first use of selection rules is in determining the symmetry classes of the states; but in a great variety of ways they may yield other information about the system and the conservation laws. *See* Quantum mechanics, Symmetry laws (physics)

For an isolated system the total angular momentum is a conserved quantity; this fact derives from a fundamental fact of nature, namely, that space is isotropic. Each state is then classifiable by angular momentum *J* and its *z* component *M* (= - *J*, -*J* + 1, …, +*J*). Angular momenta combine in a vectorial fashion. Thus, if the system makes a particle-emitting transition *J*_{1},*M*_{1} → *J*_{2},*M*_{2}, the emitted particles must carry away angular momentum (*j*, μ), where **j** = **j** _{1} - **j** _{2}. This implies that μ = *M*_{1} - *M*_{2} and that *j* takes on values *J*_{1} - *J*_{2}, *J*_{1} - *J*_{2} + 1, …, (*J*_{1} + *J*_{2}). Thus in transitions (*J* = 4 → *J* = 2) the possible *j* values comprise only 2, 3, 4, 5, 6, and, if it is also specified that *M*_{1} - *M*_{2} = ±4, only 4, 5, 6. Observe that *J*_{2} is additive. *See* Angular momentum, Quantum numbers

Another fundamental symmetry, the parity, which determines the behavior of a system (or of its description) under inversion of the coordinate axes, is conserved by the strong and electromagnetic interactions, and gives a classification of systems as even (π = +1) or odd (π = -1). Under combination the parity combines multiplicatively. Thus, if the transition above is 4^{±} → 2^{±} , it follows that *j*^{π} = 2^{-}, …, 6^{-}, while 4^{±} → 2^{±} would give *j*^{π} = 2^{+}, …, 6^{+}. The angular momentum **j** may be a combination of intrinsic spin **s** and orbital angular momentum **l** . Scalar, pseudoscalar, vector, and pseudovector particles are respectively characterized by *s*^{πs} = 0^{+}, 0^{-}, 1^{-}, 1^{+}, where π_{s} is the “intrinsic” parity, while *l* always carries π_{l} = (-1)^{l}. *See* Parity (quantum mechanics), Spin (quantum mechanics)

The isospin symmetry of the elementary particles is almost conserved, being broken by electromagnetic and weak interactions. It is described by the group SU(2), of unimodular unitary transformations in two dimensions. Since the SU(2) algebra is identical with that of the angular momentum SO(3), isospin behaves like angular momentum with its three generators **T** replacing **J** .

The isospin group is a subgroup of SU(3) which defines a more complex fundamental symmetry of the elementary particles. Two of its eight generators commute, giving two additive quantum numbers, *T*_{z} and strangeness *S*^{′} (or, equivalently, charge and hypercharge). The strangeness is conserved (Δ*S*^{′} = 0) for strong and electromagnetic, but not for weak, interactions. The selection rules and combination laws for SU(3) and its many extensions, and the quark-structure underlying them, correlate an enormous amount of information and make many predictions about the elementary particles. *See* Baryon, Elementary particle, Meson, Quarks, Unitary symmetry

A great variety of other groups have been introduced to define relevant symmetries for atoms, molecules, nuclei, and elementary particles. They all have their own selection rules, representing one aspect of the symmetries of nature.

## Selection Rules

rules that determine possible quantum transitions for such entities as atoms, molecules, atomic nuclei, and interacting elementary particles. The selection rules establish which quantum transitions are allowed and which are forbidden. Allowed transitions are those where the transition probability is great. The other transitions are either strictly forbidden (the transition probability is equal to zero) or approximately forbidden (the transition probability is small). The selection rules are accordingly divided into the strict and the approximate. When the states of a system are characterized by quantum numbers, the selection rules specify the possible changes in the numbers for a given type of transition.

The selection rules are connected with the symmetry of quantum systems—that is, the invariance of the systems’ properties under certain transformations, particularly transformations of spatial coordinates and time—and with the corresponding laws of conservation. Transitions violating the strict conservation laws, such as the laws for the energy, momentum, angular momentum, and electric charge of a closed system, are absolutely prohibited.

The strict selection rules for the quantum numbers *J* and m_{j} are very important for radiative quantum transitions between the stationary states of atoms and molecules. These numbers determine the possible values of the total angular momentum *M* and of the momentum’s projection M_{z} according to the quantization rules *M ^{2}* = ħ

*+ 1) and M*

^{2}J(J_{z}= ħm

_{J}, where is ħ Planck’s constant and

*J*and m

_{J}are integers or half-integers— m

_{J}

*= J, J*— 1,…, —

*J*. These selection rules are connected with the equivalence in space of all directions (spherical symmetry for any point) and of all directions perpendicular to a certain axis z (axial symmetry). The rules correspond to the conservation of angular momentum and of the angular momentum’s projection on the z-axis. The laws of conservation of total angular momentum and of the total angular momentum’s projection for a system consisting of microparticles and of photons that are emitted, absorbed, and scattered imply that in a quantum transition

*J*and m

_{J}can change only by 0 or ± 1 in the case of electric and magnetic dipole radiation and by 0, ±1, or ±2 in the case of electric quadrupole radiation or Raman scattering.

Another important selection rule is connected with the law of conservation of total parity for an isolated quantum system, a law violated only by weak interactions of elementary particles. The quantum states of atoms, which always have a center of symmetry, and of those molecules and crystals that have such a center are divided into even and odd with respect to space inversion, or reflection at the center of symmetry, that is, with respect to the transformation of coordinates *x’* → *— x, y’* → *—y, z’* → *— z.* In these cases, the Laporte rule for radiative quantum transitions holds: transitions between states of identical parity, that is, between even states or between odd states, are forbidden for electric dipole radiation. It is also true that transitions between states of different parity, that is, between odd and even states, are forbidden for dipole magnetic and quadrupole electric radiation and for Raman scattering. Because of these prohibitions, it is possible to observe, especially in the atomic spectra of astronomical objects, forbidden lines—lines corresponding to magnetic dipole and electric quadrupole transitions, transitions that have a very low probability in comparison with electric dipole transitions.

Also important, in addition to the exact selection rules for J and m_{J}, are approximate selection rules for the quantum numbers that determine the values of the orbital and spin angular momenta of electrons and of the projections of these momenta during dipole radiation of atoms. For example, for an atom with one outer electron, the azimuthal quantum number *l*, which determines the value of the orbital angular momentum of the electron M_{2}, M^{2}_{l} = ħ^{2}l(*l* + 1), can change by ±1; Δl = 0 is impossible, since states with identical *l* have the same parity— they are even for even l and odd for odd *l*. For complex atoms, the quantum number *L*, which determines the total orbital angular momentum of all electrons, obeys the approximate selection rule Δ *L* = 0, ±1. The quantum number *S*, which determines the total spin angular momentum of all electrons and also the multiplicity k = 2 *S* + 1, obeys the approximate selection rule *AS* = 0, which is valid if the spin-orbit interaction is neglected. Taking the interaction into account violates this last selection rule, and intercombination transitions, whose probabilities are greater the greater the atomic number of the element, appear.

Special selection rules for electronic, vibrational, and rotational molecular spectra exist for molecules, rules that are determined by the symmetry of equilibrium configurations of the molecules. In addition, for the electronic and vibrational spectra of crystals there exist selection rules that are determined by the symmetry of the crystal lattice.

In addition to the general laws of conservation of energy, momentum, and angular momentum, the physics of elementary particles has conservation laws connected with the symmetries of the fundamental interactions of particles—strong, electromagnetic, and weak interactions. The processes of the transformation of elementary particles obey the strict laws of conservation of electric charge *Q*, baryon charge *B*, and, apparently, lepton charge *L*. Strict selection rules correspond to these laws: Δ *Q =* Δ *B =* Δ *L =* 0. There also exist approximate selection rules. The selection rule for the total isotopic spin *I*, Δ *I* = 0, follows from the isotopic invariance of strong interactions. This selection rule is violated by electromagnetic and weak interactions. The selection rule for strangeness *S*, Δ *S* = 0, holds for strong and electromagnetic interactions; weak interactions occur in violation of this selection rule: ǀ Δ *S ǀ* = 1. As was noted above, in processes induced by a weak interaction the law of conservation of space parity, which is valid for all other types of interaction, is also violated. Other selection rules also exist.

M. A. EL’IASHEVICH